What to do

I like 20K bet

Nice find Colin!
Value of the advantage with 20K bet (BR2 push and BR1 lose on the penultimate hand, then BR2 bj, BR1 not winning double) is ~ 0.06x0.075x0.6=0.0027, that’s slightly better than one in four hundred plays.

But, there will be some other possible benefits, for example, when bet is lost, having 111K not 106K vs. (now new) BR1’s 125K allows us to bet 33K, which is more than twice the gap, gives us a chance for BR1 mistake giving us low with his max bet, and our double covers BR1’s max bet, so does three-bet vs. bj, but we could split to three hands (betting more than twice the gap with 106K does not). That’s still a very long shot against good player but worth trying against not so good ones.

Value of 30K bet is in BR1 receiving a blackjack. With bankroll of 176K we are forcing BR2 to win double against our push or loss. With bankroll of 161K (bj on 20K bet) and bet covering BR2 double on the last hand we are giving BR2 chance to play for a gain. Assuming that this extra chance improves BR1 chances by about 10%, it is worth ~ 0.045x0.46x0.1=0.002
(0.46 is approximate chance of BR2 winning her hand with a reasonable usage of power-chip, but not by a bj, bj would happened about 9% of times)
And 30K bets also gives us advantage when both players get bj and then we push and BR2 wins single bet while trying to swing us.
~0.045x0.09x0.05=0.0002 Hurray! Another extra chance once in 5,000 plays.

Kidding aside, the bets are virtually identical. I like 20K better for an increased chance of BR2 making betting mistake on the last hand and giving us low.

S. Yama

PS I used approximate, not exact, numbers
PPS Changed mistyped 0.36 to 0.46 -chance of winning (not bj) with power-chip for this particular situation

 

The big question

So that's 20K in preference to 30K or 25K, but how does 20K compare with 5K or 36K?

 

Favoring 30K bet again

First of all I would like to straighten bets of 20K and 30K.
Funny, the morning after I posted about them, I woke up with a thought that one important aspect was missing (I must be getting reeeally old if I have no better subjects to dream about, lol).
Whenever BR2 wins (getting to 100K or 125K with bj) BR1 double or triple win with bet of 30K moves him to one higher level of half maximum bets.
Winning double and triple bets with 20K bet bring his bankroll to respectively 171K and 191K, while with minimum of 25K bet it is 181K and 206K.
As we will see from my next post it is worth, depending on players skills, 5% to 10% better chance to advance.
It will be especially significant if BR2 makes a mistake and stands on stiff (12 to 16, occasionally 17 – 52% and 60% of times) against dealer’s upcard 2 to 6. This scenario will play out 20% of the times. Now, BR1, playing after BR2, can double being guaranteed not to bust half the times (48%), and no-bust triple bet 10%.
Significant improvement.

Next will be BR1’s advantages based on leading ½, 1, 1 ½ max bet.

S. Yama

 

small lead

Different BR1’s bets on the next-to-last hand would result in different frequencies of having bankrolls difference, from minimum lead to lock, going to the last hand. We should know what edge each lead provides to decide on right play on the next-to-last hand.
Usually considering optimum plays is a good guidance, but in this case even very good players often would play many hands less than optimally. This is due to BR1 betting in front of BR2 and often times having to choose between surrendering, playing to correlate or doubling down to limit BR2 chances of winning doubled bet –these are not basic strategy plays. Rounding and ranges instead of strict numbers for assessing edges seems more adequate.

BR2 playing correctly on the penultimate hand will use power-chip about 30% of the times, which means that in most situations on the last hand both players could use help of that rule.

BR1 HAS SMALL LEAD (for example 106K versus BR2’s 100K)

BR1 bets less than his lead.
BR2 simply advances whenever she wins big bet, with power-chip available 70% of the times; she advances in the high 50’s %.

BR1 bets 25K (half of the maximum bet)
If BR2 responses with small bet (“taking the low”) – BR1 has to push or win. BR2 chances of advancing are in low 30’s %.
If BR2 responses with maximum bet (50K) – BR2 has to make any “gain” (w/l, w/p, p/s, and p/l) or win double bet. In this scenario BR2 chances are in mid 40’s %.
If BR2 responses with bets over 6K but less than 13K she takes away from BR1 surrender but can’t catch his win even with her own double win – chances in mid 30’s %. A better bet is 16K to 18K where BR2 advances with a “gain” and both players losing and by winning double bet when BR1 wins single bet at the most – chances are in high 40’s %.

BR1 bets 50K (maximum bet)
BR2 bets over 32K have chances of advancing in mid to high 40’s %. Betting less than 38K is optimal, even though BR2’s bj is not enough to overcome BR1’s double, it prevents BR1 from surrendering, as BR2 can lock him out with her own surrender.

I think that with a small lead the best bet for BR1 is 50K, which with a good play gives BR2, at the most, chances in mid 40’s %, and have a good chance of BR1 making a mistake of taking “the low” or replying with max bet.

S. Yama

PS
Perpetual apology for long delays in my posting.